csdm - monday, october 5, 2009 - arad.pdf
TRANSCRIPT
The detectability lemma and quantum gap amplification
Dorit Aharonov1, Itai Arad1, Zeph Landau2 and Umesh Vazirani2
1School of Computer Science and Engineering,The Hebrew University,
Jerusalem, Israel
2Electrical Engineering and Computer Science,UC Berkeley,
California, USA
October 2, 2009
D. Aharonov, I. Arad, Z. Landau, U. Vazirani ( School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel, Electrical Engineering and Computer Science, UC Berkeley, California, USA)The Detectability & Gap amplification October 2, 2009 1 / 24
Quantum Hamiltonian complexity
Settings:n-variables,local constraints
Main concepts:Satisfying assignments,Reductions,Gap amplification,PCP
CSP
Settings:n quantum particleslocal quantum constrains (Hamiltonians)
Main concepts:“Satisfying” quantum states (ground states)Structure of entanglement
Condensed Matter
Hamiltonian ComplexityMain result: Quantum Cook-Levin (Kitaev ’02)
Analog of CSP results for Hamiltonians:Reductions,
completeness,(PCP ?)
Classical simulationsof Hamiltonian
systems
D. Aharonov, I. Arad, Z. Landau, U. Vazirani ( School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel, Electrical Engineering and Computer Science, UC Berkeley, California, USA)The Detectability & Gap amplification October 2, 2009 2 / 24
CSP in quantum languageMAX-3-SATSettings:
n bits: x1, . . . xn
M constraints:f(x1, . . . , xn) = C1 ∧ · · · ∧ CMCi – 3-local CNF clause
Goal: find the minimal possible # ofviolations
MAX-3-SAT – in quantum languageSettings:
Hilbert space of n qubits – 2n basisstates:|00 · · · 00〉 , |00 · · · 01〉 , . . . , |11 · · · 11〉Hamiltonian with M terms:H = Q1 + . . .+QM
Qi – 3-local “classical projections”
Goal: approximate the lowest eigenvalueof H: λ(H) = 0 or λ(H) ≥ 1.
Example:Ci = (xk ∨ x` ∨ xm)
Violating when (xk, x`, xm) = (0, 1, 0)
=⇒
00
10
00
00
|010〉
|010〉
0
0Qi = ⊗1
CSP is satisfiable ⇔ λ(H) = 0
D. Aharonov, I. Arad, Z. Landau, U. Vazirani ( School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel, Electrical Engineering and Computer Science, UC Berkeley, California, USA)The Detectability & Gap amplification October 2, 2009 3 / 24
Moving to eigenvectors and eigenvalues
Local Hamiltonian: H = Q1 +Q2 + . . . 2n × 2n matrix
Ci = (xk ∨ x` ∨ xm)
Violating when (xk, x`, xm) = (0, 1, 0)=⇒
00
10
00
00
|010〉
|010〉
0
0Qi = ⊗1
Eigenvectors: H |x1, . . . , xn〉 = (# violations) |x1, . . . , xn〉
Eigenvalue of |x1, . . . , xn〉: # of violations.
λ(H) = Lowest eigenvalue = minimal # of violations.
The problem: Decide whether λ(H) = 0 or λ(H) ≥ 1
D. Aharonov, I. Arad, Z. Landau, U. Vazirani ( School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel, Electrical Engineering and Computer Science, UC Berkeley, California, USA)The Detectability & Gap amplification October 2, 2009 4 / 24
Moving to the general quantum CSP
Quantum CSP (The Local Hamiltonian problem)
H =PMi=1Qi general local projections λ(H) = lowest eigenvalue of H
Decide: λ(H) = 0 or λ(H) ≥ 1poly(n)
Note:Qi are no longer diagonal in the standard basis,and the eigenvectors of H are superpositions:
|ψ〉 =
2n termsz | a1 |0 · · · 00〉+ a2 |0 · · · 01〉+ . . .
A different view on LH:
λ(H) = minψ〈ψ|H |ψ〉
= minψ〈ψ|
Xi
Qi |ψ〉
= minψ
Xi
〈ψ|Qi |ψ〉
0 ≤ 〈ψ|Qi |ψ〉 ≤ 1 –The energy of the constraint:how much it is violated.
(compare to 0 or 1 in the classical case)
D. Aharonov, I. Arad, Z. Landau, U. Vazirani ( School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel, Electrical Engineering and Computer Science, UC Berkeley, California, USA)The Detectability & Gap amplification October 2, 2009 5 / 24
Hamiltonian Complexity - The quantum analog of CSP
Quantum NP (QMA)
Decision problems that can be solved with a quantumwitness |ψ〉 and a polynomial quantum verifier Vx.
x ∈ L : ∃ |ψx〉 s.t. Pr[V (x, |ψx〉) = yes] ≥ 2/3 ,
x /∈ L : ∀ |φ〉 ,Pr[V (x, |φ〉) = yes] ≤ 1/3 .
x |ψx〉
V
NOYES
Quantum Cook-LevinThe Local-Hamiltonian problem is QMA-complete (Kitaev, 98).
Inclusion is the easy direction
Hardness is similar to Cook-Levin – but with some quantum twists.
D. Aharonov, I. Arad, Z. Landau, U. Vazirani ( School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel, Electrical Engineering and Computer Science, UC Berkeley, California, USA)The Detectability & Gap amplification October 2, 2009 6 / 24
Central results in Hamiltonian Complexity
Many results and techniques can be imported from CSP to Hamiltonian complexity:
Reductions using Gadgets
2-Local (even planar) Hamiltonian is QMAcomplete
Satisfaction Threshold
Lovasz local lemma
Aharonov, van Dam, Kempe, Landau,Lloyd, Regev ’04Kempe, Kitaev, Regev ’04Oliveira, Terhal ’05Najag ’07Shondhi ’09Bravyi ’09Ambainis, Kempe, Sattath ’09
But sometimes things are different:
Local Hamiltonian in 1d is QMA-complete (but1d CSP is in P).
Aharonov, Gottesman, Irani, Kempe’07
What about Quantum PCP (QPCP) ?
D. Aharonov, I. Arad, Z. Landau, U. Vazirani ( School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel, Electrical Engineering and Computer Science, UC Berkeley, California, USA)The Detectability & Gap amplification October 2, 2009 7 / 24
Quantum PCP
Two ways to formulate the QPCP conjecture:
QPCP conjecture I∀L ∈ QMA there is a quantum
verifier that has access to only q
random qubits from the witness.
Quantumreduction⇐⇒
QPCP conjecture IIDeciding whether
λ(H) = 0 or
λ(H) ≥ c ·Mis QMA-hard.
This is, however, a difficult problem
D. Aharonov, I. Arad, Z. Landau, U. Vazirani ( School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel, Electrical Engineering and Computer Science, UC Berkeley, California, USA)The Detectability & Gap amplification October 2, 2009 8 / 24
Difficulties in proving QPCP
Main Problem: no cloningThere is no quantum transformation |ψ〉 |0〉 7→ |ψ〉 |ψ〉 for all |ψ〉
No cloning was thought to prevent QECCs – but QECCs exist!
QPCP seems to require even more (QECCs are not locally decodable)
Possible implications of resolving this question:
D. Aharonov, I. Arad, Z. Landau, U. Vazirani ( School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel, Electrical Engineering and Computer Science, UC Berkeley, California, USA)The Detectability & Gap amplification October 2, 2009 9 / 24
We focus on a central ingredient in Dinur’s PCP proof, which does not require cloning:
Gap Amplification
D. Aharonov, I. Arad, Z. Landau, U. Vazirani ( School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel, Electrical Engineering and Computer Science, UC Berkeley, California, USA)The Detectability & Gap amplification October 2, 2009 10 / 24
What we want to amplify
In the Classical case
UNSATdef=
min # violationsM
Find an efficient transformation to anew CSP s.t.,
0
1
p0
1c · t · p
UNSAT
Factor t
amp’
In the Quantum case
QUNSATdef=
λ(H)
M
Find an efficient transformation to anew L.H. s.t.,
0
1
p0
1c · t · p
QUNSAT
Factor t
amp’
D. Aharonov, I. Arad, Z. Landau, U. Vazirani ( School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel, Electrical Engineering and Computer Science, UC Berkeley, California, USA)The Detectability & Gap amplification October 2, 2009 11 / 24
The probability of detecting a violation in a random constraint
Classically:
M constraints
pMviolations
p – Probability of picking a violatedconstraint
p =# of violations
M≥ UNSAT
Quantumly:
Quantum Measurement
|ψ〉Π|ψ〉
(1− Π)|ψ〉
Prob = ‖Πψ‖2
Prob = ‖(1− Π)ψ‖2
1
0
In our case, when measuring Qi:
Pr[1] = ‖Qi |ψ〉 ‖2 = 〈ψ|Qi |ψ〉= constraint’s energy
The probability of measuring a violation fora random i.
p =1
M
Xi
〈ψ|Qi |ψ〉 =λ(H)
M≥ QUNSAT
We want to amplify the probability of measuring a violation
D. Aharonov, I. Arad, Z. Landau, U. Vazirani ( School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel, Electrical Engineering and Computer Science, UC Berkeley, California, USA)The Detectability & Gap amplification October 2, 2009 12 / 24
Amplification by repitition
Classically
Pick t random constraints
Probability of at least 1 violation:
Pr = 1− (1− p)t ' t · p
New system size: M t
M constraints
pMviolations
Quantumly, this also works:
Choose t random constraintsMeasure them one after the otherAfter every measurement, the system collapses into a new state:|ψ1〉 → |ψ2〉 → · · · → |ψt〉, but always:P
i 〈ψj |Qi |ψj〉M
≥ λ(H)
M
Probability of measuring at least one violation:
Pr ≥ 1− (1− λ(H)/M)2 ' t · λ(H)/M
D. Aharonov, I. Arad, Z. Landau, U. Vazirani ( School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel, Electrical Engineering and Computer Science, UC Berkeley, California, USA)The Detectability & Gap amplification October 2, 2009 13 / 24
Amplification with expanders
Ajtai et al, Impagliazzo & Zuckerman – RP & BPP amp’
Expander graphsTaking a t-walk on a d-regular expander graph isalmost like picking t random edges.Advantage: The new system is much smaller: Mdt
(instead of M t)
satisfiededges
unsatisfiededges
C1
C2 C3 C4C5
Ce = C1 ∧ C2 ∧ C3 ∧ C4 ∧ C5
D. Aharonov, I. Arad, Z. Landau, U. Vazirani ( School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel, Electrical Engineering and Computer Science, UC Berkeley, California, USA)The Detectability & Gap amplification October 2, 2009 14 / 24
Quantum Conspiration?
Conspiration Theory
Classical correlations decay exponentially fast on anexpander, but is this also true for quantum correlations?
How do we know that the expander is “random enough”?– Can the system trick us by collapsing adversely ?
Classically: well-defined partition.
Quantumly:no assignment⇒ no partition.
Quantum World
Energy distribution
Classical World
satisfiededges
unsatisfiededges
Is there a distribution of assignments? For two constraints A,B, Pr(A = 1, B = 0)is not well defined:
Pr
h(A=1) → (B=0)
i= ‖(I −B)A |ψ〉 ‖2 6= ‖A(I −B) |ψ〉 ‖2 = Pr
h(B=0) → (A=1)
iA and B do not commute⇒ no distribution over assignments
D. Aharonov, I. Arad, Z. Landau, U. Vazirani ( School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel, Electrical Engineering and Computer Science, UC Berkeley, California, USA)The Detectability & Gap amplification October 2, 2009 15 / 24
Our Results
Quantum Gap AmplificationConsider a quantum CSP problem on a d-regular expander graph G = (V,E) with asecond largest eigvenvalue 0 < λ < 1. The edges are projections and the vertices arequdits of dimension W .
A1
A2 A3A4A5
A~e = A1 ∩ A2 ∩ A3 ∩ A4 ∩ A5
Let Gt be the hyper graph of t-walks derived from G, and for each such t-walk define aprojection into the intersection of all accepting subspaces along the walk.
Then:
QUNSAT(Gt) ≥ c(λ)K(q, d) min“t ·QUNSAT(G), 1
”D. Aharonov, I. Arad, Z. Landau, U. Vazirani ( School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel, Electrical Engineering and Computer Science, UC Berkeley, California, USA)The Detectability & Gap amplification October 2, 2009 16 / 24
Layers
layer 1layer 2layer 3layer 4
Typically, constraints can be arranged in a finite number g of layers.
In a fixed layer, all projections commute and can be measured simultaneously –there exists an underlying joint probability distribution.
With respect to one layer, we have a distribution of classical systems:
= + + + . . .
We can use the classical PCP result on each member.
There are only g layers⇒ there must be a layer with a constant energy.
But there’s a problem . . .
D. Aharonov, I. Arad, Z. Landau, U. Vazirani ( School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel, Electrical Engineering and Computer Science, UC Berkeley, California, USA)The Detectability & Gap amplification October 2, 2009 17 / 24
The problem (yet another conspiration)
It is trivial to satisfy all constraints in one fixed layer.But not simultaneously in all layers (because λ(H) > 0).
What if the violation/satisfation distribution in every layer conspires to be:
Violations
Weight
1− 1/poly(n)
1/poly(n)
0% 100%
|ψ〉 = 0.99 |no violations〉+ 0.01 |full violation〉
No amplification:In the “no violations” part there isno amplification because allconstraints are satisfied.
In the “full violations” part there isno amplification because we havereached the maximum.
We must somehow rule out this possibility if we want to show amplification
D. Aharonov, I. Arad, Z. Landau, U. Vazirani ( School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel, Electrical Engineering and Computer Science, UC Berkeley, California, USA)The Detectability & Gap amplification October 2, 2009 18 / 24
The detectability lemma
The detectability lemma (` = 0 case)Settings:⇒ Π(j) is the projection into the satisfying subspace of the j’th layer⇒ Minimal energy is ε0
def= λ(H) > 0
⇒ Projections are taken from a fixed set.
Then:‖Π(1) · · ·Π(g) |ψ〉 ‖2 ≤ 1
ε0/c+ 1' 1− ε0/c
where c is constant independent of n.
In a sequential measurent of all layers, the probability of not detecting any violations isbounded away from 1 by a constant.
CorollaryThere must be at least one layer j with a constant projection on the violating subspace
‖(I −Π(j)) |ψ〉 ‖ ≥ c′
D. Aharonov, I. Arad, Z. Landau, U. Vazirani ( School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel, Electrical Engineering and Computer Science, UC Berkeley, California, USA)The Detectability & Gap amplification October 2, 2009 19 / 24
General outline of the proofDefine
|Ω〉 def= Π(1) · · ·Π(g) |ψ〉
We wish to show ‖Ω‖2 ≤ 1ε0/c+1
.Note:
‖Ω‖2ε0 ≤ 〈Ω|H |Ω〉
We prove:
〈Ω|H |Ω〉 ≤ c`1− ‖Ω‖2
´⇓
‖Ω‖2ε0 ≤ c`1− ‖Ω‖2
´⇓
‖Ω‖2 ≤ 1
ε0/c+ 1
In the commuting case, 〈Ω|H |Ω〉 = 0 (trivial)In the general case, we show that energy contribution due to non-commuteness isexponentially decreasing.We must find a way to quantify non-commuteness.
D. Aharonov, I. Arad, Z. Landau, U. Vazirani ( School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel, Electrical Engineering and Computer Science, UC Berkeley, California, USA)The Detectability & Gap amplification October 2, 2009 20 / 24
The XY decomposition I – Pyramids
Pyramids
For every projection Q, we definea set of projections Pyr[Q]
Hpyr - The supporting Hilbertspace.
Hpyr︷ ︸︸ ︷Q
The XY decomposition
Hpyr = X ⊕ YX = common eigenvectors
Y = The rest
PX , PY = Projections to X,Y
Observations:PX , PY commute with all Q′ ∈ Pyr[Q]
‖PY`Q
Q′∈Pyr[Q]Q′´PY ‖ ≤ θ < 1
If the projections are drawn from afixed set, θ < 1 is constant!
D. Aharonov, I. Arad, Z. Landau, U. Vazirani ( School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel, Electrical Engineering and Computer Science, UC Berkeley, California, USA)The Detectability & Gap amplification October 2, 2009 21 / 24
The XY decomposition II – PonzisA Ponzi is a set of constraints from a fixed layer, whose pyramids don’t intersect
H(1)pyr︷ ︸︸ ︷ H(2)
pyr︷ ︸︸ ︷ H(3)pyr︷ ︸︸ ︷
A constant number of Ponzi’s is needed to cover all constraintsEponz – The energy operator of a Ponzi.The entire Hilbert space is decomposed in terms of XY sectors ν:
An XY sector: ν = (X,X, Y,X, Y, . . .)
Projection onto ν: Pν = PX ⊗ PX ⊗ PY ⊗ PX · · ·|ν| = No. of Y spaces
|Ω〉 =Xν
Pν |Ω〉def=Xν
|Ων〉
Eponz commutes with the decomp’:
〈Ω|Eponz |Ω〉 =Xν
〈Ων |Eponz |Ων〉
D. Aharonov, I. Arad, Z. Landau, U. Vazirani ( School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel, Electrical Engineering and Computer Science, UC Berkeley, California, USA)The Detectability & Gap amplification October 2, 2009 22 / 24
The energy of Ponzi
H(1)pyr︷ ︸︸ ︷ H(2)
pyr︷ ︸︸ ︷ H(3)pyr︷ ︸︸ ︷
ν = (X,X, Y,X, Y, . . .)|ν| = No. of Y spaces|Ω〉 def
=Pν |Ων〉
Π(1) · · ·Π(g) = D ·R |Ω〉 = D ·R |ψ〉 def= D |φ〉
|Ων〉 = PνD |φ〉 = PνDPν |φ〉 = (PνDPν) |φν〉
Exponential decay: ‖PνDPν‖ ≤ θ|ν|
Only the Y pyramids contribute to 〈Ων |Eponz |Ων〉, but their norm decaysexponentially!
〈Ων |Eponz |Ων〉 ≤ |ν|θ|ν|‖φν‖2
We obtain an upper bound: 〈Ω|Eponz |Ω〉 ≤P∞s=0 sθ
s.
D. Aharonov, I. Arad, Z. Landau, U. Vazirani ( School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel, Electrical Engineering and Computer Science, UC Berkeley, California, USA)The Detectability & Gap amplification October 2, 2009 23 / 24
The energy of Ponzi
H(1)pyr︷ ︸︸ ︷ H(2)
pyr︷ ︸︸ ︷ H(3)pyr︷ ︸︸ ︷
R
D
ν = (X,X, Y,X, Y, . . .)|ν| = No. of Y spaces|Ω〉 def
=Pν |Ων〉
Π(1) · · ·Π(g) = D ·R |Ω〉 = D ·R |ψ〉 def= D |φ〉
|Ων〉 = PνD |φ〉 = PνDPν |φ〉 = (PνDPν) |φν〉
Exponential decay: ‖PνDPν‖ ≤ θ|ν|
Only the Y pyramids contribute to 〈Ων |Eponz |Ων〉, but their norm decaysexponentially!
〈Ων |Eponz |Ων〉 ≤ |ν|θ|ν|‖φν‖2
We obtain an upper bound: 〈Ω|Eponz |Ω〉 ≤P∞s=0 sθ
s.
D. Aharonov, I. Arad, Z. Landau, U. Vazirani ( School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel, Electrical Engineering and Computer Science, UC Berkeley, California, USA)The Detectability & Gap amplification October 2, 2009 23 / 24
Open questions
The quantum amplification lemma is trivial to prove when all the projectionscommute (one does not need the detectability lemma). Still it is not clear:
I if the quantum PCP can be proved for commuting HamiltoniansI what is the complexity of this special class? is it NP? QMA ?
Can the XY decomposition or the detectability lemma be used elsewhere tohandle the non-commuteness of the LH problem ?
Are there any interesting implications of the detectability lemma and the XYdecomposition to solid-state physics?
Currently, the detectability lemma allows us an RP type of amplification (one-sidederrors). Can it be generalized to prove a BPP type of amplification (two-sidederrors) ?
What is the right definition of quantum PCP? (one-sided errors/two sided errors?)
Can we prove a quantum PCP with exponential witness ?
If there is no quantum PCP theorem, then what is the complexity of approximatingQUNSAT(H) up to a constant? It must be NP-hard - but is it inside NP?
D. Aharonov, I. Arad, Z. Landau, U. Vazirani ( School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel, Electrical Engineering and Computer Science, UC Berkeley, California, USA)The Detectability & Gap amplification October 2, 2009 24 / 24
Open questions
The quantum amplification lemma is trivial to prove when all the projectionscommute (one does not need the detectability lemma). Still it is not clear:
I if the quantum PCP can be proved for commuting HamiltoniansI what is the complexity of this special class? is it NP? QMA ?
Can the XY decomposition or the detectability lemma be used elsewhere tohandle the non-commuteness of the LH problem ?
Are there any interesting implications of the detectability lemma and the XYdecomposition to solid-state physics?
Currently, the detectability lemma allows us an RP type of amplification (one-sidederrors). Can it be generalized to prove a BPP type of amplification (two-sidederrors) ?
What is the right definition of quantum PCP? (one-sided errors/two sided errors?)
Can we prove a quantum PCP with exponential witness ?
If there is no quantum PCP theorem, then what is the complexity of approximatingQUNSAT(H) up to a constant? It must be NP-hard - but is it inside NP?
D. Aharonov, I. Arad, Z. Landau, U. Vazirani ( School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel, Electrical Engineering and Computer Science, UC Berkeley, California, USA)The Detectability & Gap amplification October 2, 2009 24 / 24
Open questions
The quantum amplification lemma is trivial to prove when all the projectionscommute (one does not need the detectability lemma). Still it is not clear:
I if the quantum PCP can be proved for commuting HamiltoniansI what is the complexity of this special class? is it NP? QMA ?
Can the XY decomposition or the detectability lemma be used elsewhere tohandle the non-commuteness of the LH problem ?
Are there any interesting implications of the detectability lemma and the XYdecomposition to solid-state physics?
Currently, the detectability lemma allows us an RP type of amplification (one-sidederrors). Can it be generalized to prove a BPP type of amplification (two-sidederrors) ?
What is the right definition of quantum PCP? (one-sided errors/two sided errors?)
Can we prove a quantum PCP with exponential witness ?
If there is no quantum PCP theorem, then what is the complexity of approximatingQUNSAT(H) up to a constant? It must be NP-hard - but is it inside NP?
D. Aharonov, I. Arad, Z. Landau, U. Vazirani ( School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel, Electrical Engineering and Computer Science, UC Berkeley, California, USA)The Detectability & Gap amplification October 2, 2009 24 / 24
Open questions
The quantum amplification lemma is trivial to prove when all the projectionscommute (one does not need the detectability lemma). Still it is not clear:
I if the quantum PCP can be proved for commuting HamiltoniansI what is the complexity of this special class? is it NP? QMA ?
Can the XY decomposition or the detectability lemma be used elsewhere tohandle the non-commuteness of the LH problem ?
Are there any interesting implications of the detectability lemma and the XYdecomposition to solid-state physics?
Currently, the detectability lemma allows us an RP type of amplification (one-sidederrors). Can it be generalized to prove a BPP type of amplification (two-sidederrors) ?
What is the right definition of quantum PCP? (one-sided errors/two sided errors?)
Can we prove a quantum PCP with exponential witness ?
If there is no quantum PCP theorem, then what is the complexity of approximatingQUNSAT(H) up to a constant? It must be NP-hard - but is it inside NP?
D. Aharonov, I. Arad, Z. Landau, U. Vazirani ( School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel, Electrical Engineering and Computer Science, UC Berkeley, California, USA)The Detectability & Gap amplification October 2, 2009 24 / 24
Open questions
The quantum amplification lemma is trivial to prove when all the projectionscommute (one does not need the detectability lemma). Still it is not clear:
I if the quantum PCP can be proved for commuting HamiltoniansI what is the complexity of this special class? is it NP? QMA ?
Can the XY decomposition or the detectability lemma be used elsewhere tohandle the non-commuteness of the LH problem ?
Are there any interesting implications of the detectability lemma and the XYdecomposition to solid-state physics?
Currently, the detectability lemma allows us an RP type of amplification (one-sidederrors). Can it be generalized to prove a BPP type of amplification (two-sidederrors) ?
What is the right definition of quantum PCP? (one-sided errors/two sided errors?)
Can we prove a quantum PCP with exponential witness ?
If there is no quantum PCP theorem, then what is the complexity of approximatingQUNSAT(H) up to a constant? It must be NP-hard - but is it inside NP?
D. Aharonov, I. Arad, Z. Landau, U. Vazirani ( School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel, Electrical Engineering and Computer Science, UC Berkeley, California, USA)The Detectability & Gap amplification October 2, 2009 24 / 24
Open questions
The quantum amplification lemma is trivial to prove when all the projectionscommute (one does not need the detectability lemma). Still it is not clear:
I if the quantum PCP can be proved for commuting HamiltoniansI what is the complexity of this special class? is it NP? QMA ?
Can the XY decomposition or the detectability lemma be used elsewhere tohandle the non-commuteness of the LH problem ?
Are there any interesting implications of the detectability lemma and the XYdecomposition to solid-state physics?
Currently, the detectability lemma allows us an RP type of amplification (one-sidederrors). Can it be generalized to prove a BPP type of amplification (two-sidederrors) ?
What is the right definition of quantum PCP? (one-sided errors/two sided errors?)
Can we prove a quantum PCP with exponential witness ?
If there is no quantum PCP theorem, then what is the complexity of approximatingQUNSAT(H) up to a constant? It must be NP-hard - but is it inside NP?
D. Aharonov, I. Arad, Z. Landau, U. Vazirani ( School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel, Electrical Engineering and Computer Science, UC Berkeley, California, USA)The Detectability & Gap amplification October 2, 2009 24 / 24
Open questions
The quantum amplification lemma is trivial to prove when all the projectionscommute (one does not need the detectability lemma). Still it is not clear:
I if the quantum PCP can be proved for commuting HamiltoniansI what is the complexity of this special class? is it NP? QMA ?
Can the XY decomposition or the detectability lemma be used elsewhere tohandle the non-commuteness of the LH problem ?
Are there any interesting implications of the detectability lemma and the XYdecomposition to solid-state physics?
Currently, the detectability lemma allows us an RP type of amplification (one-sidederrors). Can it be generalized to prove a BPP type of amplification (two-sidederrors) ?
What is the right definition of quantum PCP? (one-sided errors/two sided errors?)
Can we prove a quantum PCP with exponential witness ?
If there is no quantum PCP theorem, then what is the complexity of approximatingQUNSAT(H) up to a constant? It must be NP-hard - but is it inside NP?
D. Aharonov, I. Arad, Z. Landau, U. Vazirani ( School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel, Electrical Engineering and Computer Science, UC Berkeley, California, USA)The Detectability & Gap amplification October 2, 2009 24 / 24
Open questions
The quantum amplification lemma is trivial to prove when all the projectionscommute (one does not need the detectability lemma). Still it is not clear:
I if the quantum PCP can be proved for commuting HamiltoniansI what is the complexity of this special class? is it NP? QMA ?
Can the XY decomposition or the detectability lemma be used elsewhere tohandle the non-commuteness of the LH problem ?
Are there any interesting implications of the detectability lemma and the XYdecomposition to solid-state physics?
Currently, the detectability lemma allows us an RP type of amplification (one-sidederrors). Can it be generalized to prove a BPP type of amplification (two-sidederrors) ?
What is the right definition of quantum PCP? (one-sided errors/two sided errors?)
Can we prove a quantum PCP with exponential witness ?
If there is no quantum PCP theorem, then what is the complexity of approximatingQUNSAT(H) up to a constant? It must be NP-hard - but is it inside NP?
D. Aharonov, I. Arad, Z. Landau, U. Vazirani ( School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel, Electrical Engineering and Computer Science, UC Berkeley, California, USA)The Detectability & Gap amplification October 2, 2009 24 / 24